"Aye aye, sir" is a phrase commonly heard present day in naval language. It is derived from a duplicate of the word "aye" which came into the English language in the late 16th century and early 17th century, meaning "Yes; even so.".
Description
It was common in dialect and is the formal word for voting "yes" in the English House of Commons.
Its most common use is as a naval response indicating that an order has been received, is understood, and will be carried out immediately. It differs from "yes", which, in standard usage, could mean simple agreement without any intention to act. In naval custom, a reply of "yes sir" would indicate agreement to a statement that was not understood as an order or a requirement to do anything. The alternatives of "aye aye sir" and "yes sir" would allow any misunderstanding to be corrected at once. This might be a matter of life and death for a ship at sea.
Basically, it means that the speaker understands and will obey a direct order.
Aye = Yes
Aye Aye = I understand, and will comply.
In other words, it's the naval equivalent to the aeronautical difference between "Roger" and "Wilco."